Mihir

iiT,BhuBaneswar

Quantum Mechanical Wave Function gives all information about a given system.

For a Hydrogen Atom, We can solve Schrodinger Equation exactly and determine the allowed energy state of the System.

It is difficult to solve Schrodinger Equation for N- body System. We must involve some approximation for the solution .

. It is a method of obtaining an approximate solution to Schrodinger Equation for many body system. . It is Primarily a theory of Electronic ground state structure in terms of electronic density distribution n(r).

The Time independent Schrodinger Equation H𝞇𝞇 𝑥𝑥1,𝑥𝑥2, … . . 𝑥𝑥𝑁𝑁.𝑅𝑅1.𝑅𝑅2 … . .𝑅𝑅𝑀𝑀 = 𝐸𝐸𝐸𝐸 (𝑥𝑥1,𝑥𝑥2, . 𝑥𝑥𝑁𝑁,𝑅𝑅1 … …𝑅𝑅𝑀𝑀)

Where H is the Hamiltonian for a system consisting of M nuclei and N electrons.

𝐻𝐻 = −1

2𝑚𝑚𝑒𝑒�𝞩𝞩𝑖𝑖2𝑁𝑁

𝑖𝑖=1

−1

2𝑚𝑚𝑛𝑛�𝞩𝞩𝑗𝑗2𝑀𝑀

𝑗𝑗=1

+12�𝑁𝑁

𝑖𝑖=1

�𝑒𝑒2

𝑟𝑟𝑖𝑖 − 𝑟𝑟𝑘𝑘

𝑁𝑁

𝑘𝑘=1,𝑖𝑖≠𝑘𝑘

−��𝑍𝑍𝑗𝑗𝑒𝑒2

|𝑟𝑟𝑖𝑖 − 𝑟𝑟𝑘𝑘|+

12�𝑀𝑀

𝑗𝑗=1

�𝑍𝑍𝑗𝑗𝑍𝑍𝑤𝑤𝑒𝑒2

|𝑟𝑟𝑗𝑗 − 𝑟𝑟𝑤𝑤|

𝑀𝑀

𝑤𝑤=1,𝑤𝑤≠𝑗𝑗

𝑀𝑀

𝑗𝑗=1

𝑁𝑁

𝑖𝑖=1

𝐻𝐻 = 𝑇𝑇𝑒𝑒 + 𝑇𝑇𝑛𝑛 + 𝑉𝑉𝑒𝑒𝑒𝑒 + 𝑉𝑉𝑒𝑒𝑛𝑛 + 𝑉𝑉𝑛𝑛𝑛𝑛

First two terms of the above equation are Kinetic Energy of Electrons and nuclei respectively. Third term is electron- electron Potential Energy. Fourth term is the Potential energy between electrons and nuclei. Last term is nuclei-nuclei potential energy.

Born-oppenheiMer approxiMaTion

𝑚𝑚𝑛𝑛 ≫ 𝑚𝑚𝑒𝑒 Nuclei are much slower than electron. Hence Nuclei Kinetic Energy is zero. 𝑇𝑇𝑛𝑛= 0 Nuclei-Nuclei Interaction is Constant. 𝑉𝑉𝑛𝑛𝑛𝑛= 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 Hence we can eliminate the term 𝑇𝑇𝑛𝑛 𝐶𝐶𝐶𝐶𝑎𝑎 𝑉𝑉𝑛𝑛𝑛𝑛 from the Hamiltonian. Hence New Hamiltonian for many body system

𝐻𝐻 = 𝑇𝑇𝑒𝑒 + 𝑉𝑉𝑒𝑒𝑒𝑒 + 𝑉𝑉𝑒𝑒𝑛𝑛

All ground state properties are determined by ground state density n(r).

The total energy of a many body system is a unique functional of electron density.

Density 𝐶𝐶 𝑟𝑟 = ∫𝑎𝑎3𝑟𝑟2 ∫𝑎𝑎3𝑟𝑟3 … …∫𝑎𝑎3𝑟𝑟𝑁𝑁 |𝞇𝞇 𝑟𝑟, 𝑟𝑟2, … 𝑟𝑟𝑁𝑁 |2 n(r) uniquely determines 𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒. 𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 = ∫ 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑟𝑟 𝐶𝐶 𝑟𝑟 𝑎𝑎𝑟𝑟 The total energy functional can be written as 𝐸𝐸 𝐶𝐶 = 𝐹𝐹 𝐶𝐶 + ∫𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑟𝑟 𝐶𝐶 𝑟𝑟 𝑎𝑎𝑟𝑟

Universal function F(n) is Independent of External potential but it is unknown.

INPUT OUTPUT

INCAR POSCAR POTCAR KPOINTS

OUTCAR OSCZICAR CONTCAR CHGCAR EIGENVAL WAVECAR PROCAR XDATCAR DOSCAR

INCAR: Central Input file. :Different parameters for different properties. . POSCAR: Position of Ions : Lattice Constant .POTCAR : Psuedopotential from VASP .KPOINTS: Kpoint meshes

For Binding Energy calculation, I have to calculate single point Energy calculation.

I take Oxygen as an example. Here I do spin Polarised calculation. For single O atom, Input files are INCAR,

POSCAR, POTCAR, KPOINTS. For O2 dimer, only POSCAR file is modified. B.E =E(O2)-2*E(O)

INCAR POSCAR

KPOINTS

POTCAR

FREE Energy value for O Atom

Here all the Input files are same as those for Oxygen Atom.

Only POSCAR File is slightly changed. Here No. of atoms is 2 Hence two co-ordinate points are required for

describing the position of atoms in the molecule.

Energy of Oxygen Molecule

Binding Energy = E (O2)-2*E(O) = -9.83466139+3.96809746 = -5.86656393eV

Here I choose Cl2 dimer. Input files : INCAR,POSCAR, POTCAR, KPOINTS Open POSCAR file

Cl2 Molecule ! Comment line 1.00 !Length Unit in Angstrom 10.0 0.00 0.00 0.00 10.0 0.00 0.00 0.00 10.0 2 Cartesian 0.00 0.00 0.00 1.97 0.00 0.00

Lattice Vectors

No. of Atoms

Position of Atoms

Now edit the POSCAR file and Change X-Co-ordinate of second Cl atom to 1.99A

Save the file and run VASP Record Energy value. Repeat the Calculation for successive increase in bond length up

to 2.07A and record corresponding Energy values. Plot the graph between Energy and bond Length. The Length corresponds to minimum energy is the equilibrium

bond length.

Here I choose Cu atom in FCC phase. Keep all INPUT files (INCAR,POSCAR,POTCAR,KPOINTS) POSCAR File

Cu 3.55 0.5 0.5 0.0 0.0 0.5 0.5 0.5 0.0 0.5 1 Cartesian 0 0 0

Lattice vector in FCC System

Edit POSCAR file and Change Lattice Constant to 3.60 A Save the file and run the job and record energy Repeat the calculation by successive increase in length up to 3.70A

and record all corresponding Energy values. Plot the graph between energy with lattice constant. The lattice constant corresponds to minimum energy is actual

lattice constant.

The silicon Crystal structure is FCC . FCC primitive unit cell with 2 atoms in the

unit cell. For band structure calculation first run self-

consistency calculation to get the charge density.

Then fix the charge density and run a non self consistency calculation for desired K points to get band structure.

INCAR for SCF Calculation

For SCF calculation ICHARG =2

For GGA psuedopotential

POSCAR

For FCC, no. of atoms per unit cell

KPOINTS

For Automatic Mesh Generation

POTCAR file is provided by VASP Then run VASP After job is completed, we get charge density in

CHGCAR file To get band structure, run non-self consistence

calculation for each desired k points.

Input files are INCAR, POSCAR, POTCAR, KPOINTS, CHGCAR.

INCAR and KPOINTS files will be modified. In INCAR file, we change ICHARG=11 We modify KPOINTS file, to specify along

some high symmetry direction to calculate energy.

Reciprocal space

Along each line, 10 kpoints are calculated.

Run VASP The above KPOINTS file instructs vasp to

calculate the Energy at each k point between L point and Gamma point, gamma point and K point.

Along each line 10 Kpoints are calculated. In output, we find EIGENVAL file. From this we find all information to plot band

structure.

EIGENVAL file

Kpoints

Energy of 8 bands at that particular energy point

Since energy of bands at each k points, we get information about band structure.

I Plot band structure using MATLAB.

Band Structure of Silicon

Graphene is a 2-dimensional crystalline allotrope of carbon.

In graphene, Carbon atoms are densely packed in a regular hexagonal pattern.

It is one atom thick layer of graphite. Input files INCAR, POSCAR, POTCAR,

KPOINTS

INCAR POSCAR KPOINTS

INPUT files for SCF Calculation for graphene

POTCAR

KPOINTS for Graphene Along high symmetry line

Band Structure of Graphene

W.Kohn. L.J.Sham, physical rev. 140,A1133(1965)

H.J. Monkhrost and J.Pack, Phys. Rev.B 13,5188(1976)

W.Kohn, A.D.Becke,R.G.Parr,J.Phys.Chem.1996, 100, 12974-12980